Structure a family of three-step with-memory methods for solving nonlinear equations and their dynamics

In this work, we will first propose an optimal three-step without-memory method for solving nonlinear equations. Then, by introducing the self-accelerating parameters, the with-memory-methods have been built. They have a fifty-nine percentage improvement in the convergence order. The proposed methods have not the problems of calculating the function derivative. We use these Steffensen- type methods to solve nonlinear equations with simple zeroes with the appropri- ate initial approximation of the root. we have solved a few nonlinear problems to justify the theoretical study. Finally, are described the dynamics of the with- memory method for complex polynomials of degree two.